Sen. Tom Coburn (R., Okla.) is quietly circulating his own proposal for expanding gun-sale background checks, a sign that the conservative lawmaker may be trying to build support outside of long-running bipartisan negotiations on the issue.

After struggling for months to reach a deal with Democrats on how to broaden background checks, Mr. Coburn has written a 43-page draft of a bill that wouldn’t require private gun sellers to keep a paper record of the sale, according to a copy of the proposal.

Dropping the record-keeping requirement would make it difficult for Democrats to support the bill, according to individuals familiar with the negotiations. Democrats have argued there is no way to enforce the checks without some form of documentation.

Mr. Coburn has been working for weeks to reach a deal with Democratic Sens. Joe Manchin of West Virginia and Charles Schumer of New York, as well as Sen. Mark Kirk (R., Ill.) on the issue.

“Sen. Manchin continues to talk to all his colleagues,” a spokesman for Mr. Manchin said Friday.

A spokesman for Mr. Coburn said the document did not represent the current state of the negotiations.

“It is not accurate. It doesn’t reflect the status of negotiations,” Mr. Coburn’s spokesman said. “Talks are ongoing.”

However, the discussions between Mr. Coburn and Mr. Manchin have stalled, with neither side willing to yield on the record-keeping requirement, and Mr. Manchin started talks this week with Sen. Pat Toomey (R., Pa.), according to an aide familiar with the negotiations.

“The best hope we have right now of a bipartisan compromise is the discussion going on between Manchin and Toomey,” a Senate Democratic aide said Friday. “There is real potential there.”

In a blitz of town hall meetings in Oklahoma this week, Mr. Coburn assured his constituents that any bill to expand background checks to private sales would not generate any paper records. Currently, only licensed dealers must conduct background checks of potential buyers and keep a record of the sale. While he mentioned his negotiations, Mr. Coburn didn’t emphasize his work with Democrats, but did note his sway with other Senate Republicans.

“The only thing that can get through Congress is what I just described,” Mr. Coburn said at a town hall meeting in El Reno, Okla. “I have 45 people who will follow me absolutely in the U.S. Senate on what I say on this issue.”

The Senate had been expected to take up gun legislation shortly after recess, but an aide to Democratic leaders said this week it may be delayed to give lawmakers time to work out a compromise on the background checks. If reached, that would become the most ambitious piece of legislation included in the main gun-control bill. The Senate will vote on other measures that are not expected to pass, including a ban on semi-automatic rifles known as assault weapons.

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